Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants

By Boris Dubrovin, You-jin Zhang. 2001

392 citations recorded by the source corpus.

We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter. Our aim is to embed the theory of Gromov - Witten invariants of all genera into the theory of integrable systems. The project is focused at describing normal forms of the PDEs and their local bihamiltonian structures satisfying certain simple axioms. A Frobenius manifold or its degeneration is associated to every bihamiltonian structure of our type. The main result is a universal loop equation on the jet space of a semisimple Frobenius manifold that can be used for perturbative reconstruction of the integrable hierarchy. We show that first few terms of the perturbative expansion correctly reproduce the universal identities between intersection numbers of Gromov - Witten classes and their descendents.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b98ca3d77401abdfc63417548e70bff0fc758020

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

ON ALMOST DUALITY FOR FROBENIUS MANIFOLDS (2003, methodology): “sis given by the ﬂat coordinates with good behaviour at λ= ∞. Recall that this Gram matrix was used in the free ﬁeld realization of the Virasoro operators associated with the given Frobenius manifold [19]. The general formula (4.70) was used in [19] for regularization of the Virasoro operators in th”
Hirota quadratic equations for the extended Toda hierarchy (2005, background): “ied some partial result of A. Okounkov – R. Pandharipande [OP1] together with the Virasoro constraints for CP1. The paper [DZZ] by B. Dubrovin – Y. Zhang contains a diﬀerent proof based on the theory [DZ] of integrable hierarchies associated to Frobenius structures and also uses the Virasoro constra”
The jet-space of a Frobenius manifold and higher-genus Gromov-Witten invariants (2002, methodology): “Of course, it may be easily proved in this case by direct methods from the definition of the Gelfand-Dikii polynomials.”
Matrix factorizations and Cohomological Field Theories (2011, methodology): “ion of quantum cohomology, a certain deformation of the cohomology ring of X. Through Frobenius manifolds CohFTs are related to integrable hierarchies of systems of partial diﬀerential equations (see [12]). The ﬁrst example of CohFTs besides GW-theory was provided by the theory of r-spin curves cons”
The spaces of Laurent polynomials, Gromov-Witten theory of ℙ1-orbifolds and integrable hierarchies (2008, background): “In this paper we prove that if M ¼ Mk;m is the space of Laurent polynomials in one variable, then the Fourier coe‰cients of DM and AMt satisfy an infinite system of quadratic relations.”
The Extended Toda Hierarchy (2003, background): “We leave the proof as an exercise for the reader.”